Let’s go step by step to complete the tasks assigned regarding Jerome's energy drink.
Step 1: Finding the part-to-part and part-to-whole ratios
1. What is the ratio of water to drink mix?
To find the ratio of water to drink mix, we divide the number of cups of water by the number of cups of drink mix:
- Water: 8 cups
- Drink mix: 3 cups
\[ \text{Ratio of water to drink mix} = \frac{8}{3} \quad \text{(which can be written as 8:3)} \]
2. What is the ratio of drink mix to water?
To find the ratio of drink mix to water, we swap the values: \[ \text{Ratio of drink mix to water} = \frac{3}{8} \quad \text{(which can be written as 3:8)} \]
3. What is the ratio of drink mix to mixed energy drink?
First, calculate the total amount of mixed energy drink: \[ \text{Total} = \text{Water} + \text{Drink mix} = 8 + 3 = 11 \text{ cups} \]
Now, we find the ratio of drink mix to the total mixed energy drink: \[ \text{Ratio of drink mix to mixed energy drink} = \frac{3}{11} \quad \text{(which can be written as 3:11)} \]
4. What is the ratio of water to mixed energy drink?
We find the ratio of water to the total mixed energy drink: \[ \text{Ratio of water to mixed energy drink} = \frac{8}{11} \quad \text{(which can be written as 8:11)} \]
Step 2: Testing Jerome’s brother’s approach
Jerome's brother suggested adding 1 cup of water for every 1 cup of drink mix he added. Let’s complete the table.
| Original Ratio | +1 (Drink Mix) | +2 (Drink Mix) | +3 (Drink Mix) | +4 (Drink Mix) | |----------------|------------------|------------------|------------------|------------------| | Drink Mix (cups) | 3 | 4 | 5 | 6 | | Water (cups) | 8 | 9 | 10 | 11 |
b) Is Jerome’s brother correct? Explain why or why not.
Answer: Jerome’s brother is not correct. By the way he suggests adding water and drink mix, the ratio of water to drink mix changes. Originally, the ratio of water to drink mix is 8:3. If Jerome adds 1 cup of water for each additional cup of drink mix:
- For 4 cups of drink mix, the amount of water would be 9 cups, leading to a new ratio of: \[ \text{Ratio of water to drink mix} = \frac{9}{4} \quad \text{(which is 9:4)} \]
This means the ratio of water to drink mix is now 9:4, which is different from the original ratio of 8:3. Therefore, the original ratio is not maintained.
Step 3: Testing Jerome’s sister’s approach
Jerome's sister suggested multiplying both the amount of water and drink mix by the same number. Let’s complete the table using a multiplier.
| Original Ratio | ×1 | ×2 | ×3 | ×4 | |----------------|----|----|----|----| | Drink Mix (cups) | 3 | 6 | 9 | 12 | | Water (cups) | 8 | 16 | 24 | 32 |
Is Jerome’s sister correct? Explain why or why not.
Answer: Jerome’s sister is correct. By multiplying both the amount of drink mix and the amount of water by the same number, the ratio remains the same. For example:
- When multiplied by 2: \[ \text{New Water} = 8 \times 2 = 16 \quad \text{and} \quad \text{New Drink Mix} = 3 \times 2 = 6 \] The ratio remains: \[ \frac{16}{6} = \frac{8}{3} \quad \text{(which is still 8:3)} \] This shows that the original ratio is maintained.
Step 4: Calculating the right recipe for Jerome
How many cups of energy drink does Jerome need to make?
Twenty-two people each want 2 cups of energy drink: \[ \text{Total cups needed} = 22 \times 2 = 44 \text{ cups} \]
To maintain Jerome's original ratio of water to drink mix (8:3), we need to find out how many parts there are in the total energy drink:
- Original total parts = 8 + 3 = 11 parts.
To find the amount for a single part: \[ \text{Amount per part} = \frac{44 \text{ cups}}{11} = 4 \text{ cups per part} \]
Now, calculate how much water and drink mix Jerome needs:
- Water: \[ \text{Water} = 8 \text{ parts} \times 4 \text{ cups/part} = 32 \text{ cups} \]
- Drink Mix: \[ \text{Drink Mix} = 3 \text{ parts} \times 4 \text{ cups/part} = 12 \text{ cups} \]
Conclusion and Summary Table
| Ingredient | Cups Needed | |----------------|--------------| | Water | 32 | | Drink Mix | 12 |
To make enough energy drink for all the runners, Jerome should use 32 cups of water and 12 cups of drink mix. This keeps the original ratio intact while satisfying the demand for 44 cups of energy drink.